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All sourcesNASA NTRSDOE OSTI
NASA NTRS2007

Nonlinear Optics and Applications

Nonlinear optics is the result of laser beam interaction with materials and started with the advent of lasers in the early 1960s. The field is growing daily and plays a major role in emerging photonic technology. Nonlinear optics play a major role in many of the optical applications such as optical signal processing, optical computers, ultrafast switches, ultra-short pulsed lasers, sensors, laser amplifiers, and many others. This special review volume on Nonlinear Optics and Applications is intended for those who want to be aware of the most recent technology. This book presents a survey of the recent advances of nonlinear optical applications. Emphasis will be on novel devices and materials, switching technology, optical computing, and important experimental results. Recent developments in topics which are of historical interest to researchers, and in the same time of potential use in the fields of all-optical communication and computing technologies, are also included. Additionally, a few new related topics which might provoke discussion are presented. The book includes chapters on nonlinear optics and applications; the nonlinear Schrodinger and associated equations that model spatio-temporal propagation; the supercontinuum light source; wideband ultrashort pulse fiber laser sources; lattice fabrication as well as their linear and nonlinear light guiding properties; the second-order EO effect (Pockels), the third-order (Kerr) and thermo-optical effects in optical waveguides and their applications in optical communication; and, the effect of magnetic field and its role in nonlinear optics, among other chapters.

Abdeldayem, Hossin A.↗
NASA NTRSDate not supplied

The PCM is Dead! Long Live O-PCM!

The ability to reconfigure the optical behavior of a device enables free-space applications ranging from imaging to sensing and signal control. Such optical devices can be compacted via meta-surfaces, patterned structures with feature sizes below the incident wavelength. Leveraging geometry in addition to material properties and CMOS fabrication techniques has allowed meta-surfaces for lenses, holograms, beam steerers and more. To incorporate multiple optical functions into one device, various methods of device control have been implemented, such as stretching of flexible substrates, tuning the refractive index of the comprising meta-atoms via the electro-optic or the thermo-optic effects, phase transition materials such as VO2 and more. Chalcogenide glasses used as optical phase change materials, such as Ge¬2Sb2Te5 (GST), have gained increased traction in the optics community for potential use in the near infrared (NIR) and mid infrared (MIR) bands, including the telecom bands. Various chalcogenides such as Sb2Se3, Sb2S3, Ge2Sb¬2Se5 and Ge2Sb2Se4Te (GSST) have been investigated due to their broad NIR or MIR transparency window and large changes in refractive index. In their amorphous phase, these materials usually display a lower refractive index and low absorption when compared to their crystalline state which display a higher refractive index and typically larger extinction coefficients. The amorphous-crystalline reversible switching can be done via fast melt-quenching thermal processes triggered by laser or electrical impulses, relying on a substrate as a heat sink. The potential of PCMs in photonic devices can be limited by intrinsic material limitations as well as by device fabrication issues. To explore the cyclability of GSST, a PCM with large refractive index contrast and on-chip electrothermal switching on a silicon-on-insulator platform has been done to analyze potential failure mechanisms from both a material and device perspective. A brief outline of the instrumentation and phase change contrast analysis is provided. Dewetting of the PCM, delamination of and damage in the PECVD SiNx protective layer, elemental migration in the PCM and optical contrast decay have been observed in cycled GSST devices. Guidelines for device performance improvement are proposed, and an improved design with larger endurance is shown in progress.

reconfigurable photonics↗
NASA NTRS1999

Overview of Photonic Materials for Application in Space Environments

Future space systems will he based on components evolving from the development and refinement of new and existing photonic materials. Optically based sensors, inertial guidance, tracking systems, communications, diagnostics, imaging and high speed optical processing are but a few of the applications expected to widely utilize photonic materials. The response of these materials to space environment effects (SEE) such as spacecraft charging, orbital debris, atomic oxygen, ultraviolet irradiation, temperature and ionizing radiation will be paramount to ensuring successful space applications. The intent of this paper is to, address the latter two environments via a succinct comparison of the known sensitivities of selected photonic materials to the temperature and ionizing radiation conditions found in space and enhanced space environments Delineation of the known temperature and radiation induced responses in LiNbO3, AlGaN, AlGsAs,TeO2, Si:Ge, and several organic polymers are presented. Photonic materials are realizing rapid transition into applications for many proposed space components and systems including: optical interconnects, optical gyros, waveguide and spatial light modulators, light emitting diodes, lasers, optical fibers and fiber optic amplifiers. Changes to material parameters such as electrooptic coefficients, absorption coefficients, polarization, conductivity, coupling coefficients, diffraction efficiencies, and other pertinent material properties examined for thermo-optic and radiation induced effect. Conclusions and recommendations provide the reader with an understanding of the limitations or attributes of material choices for specific applications.

Taylor, E. W.↗
NASA NTRS2004

Space Environmental Effects on Candidate Solar Sail Materials

The National Aeronautics and Space Administration's (NASA) Marshall Space Flight Center (MSFC) continues research into the utilization of photonic materials for spacecraft propulsion. Spacecraft propulsion, using photonic materials, will be achieved using a solar sail. A solar sail operates on the principle that photons, originating from the sun, impart pressure to the sail and therefore provide a source for spacecraft propulsion. The pressure imparted ot a solar sail can be increased, up to a factor of two, if the sun-facing surface is perfectly reflective. Therefore, these solar sails are generally composed of a highly reflective metallic sun-facing layer, a thin polymeric substrate and occasionally a highly emissive back surface. Near term solar sail propelled science missions are targeting the Lagrange point 1 (L1) as well as locations sunward of L1 as destinations. These near term missions include the Solar Polar Imager and the L1 Diamond. The Environmental Effects Group at NASA's Marshall Space Flight Center (MSFC) continues to actively characterize solar sail material in preparation for these near term solar sail missions. Previous investigations indicated that space environmental effects on sail material thermo-optical properties were minimal and would not significantly affect the propulsion efficiency of the sail. These investigations also indicated that the sail material mechanical stability degrades with increasing radiation exposure. This paper will further quantify the effect of space environmental exposure on the mechanical properties of candidate sail materials. Candidate sail materials for these missions include Aluminum coated Mylar, Teonex, and CP1 (Colorless Polyimide). These materials were subjected to uniform radiation doses of electrons and protons in individual exposures sequences. Dose values ranged from 100 Mrads to over 5 Grads. The engineering performance property responses of thermo-optical and mechanical properties were characterized. The contribution of Near Ultraviolet (NUV) radiation combined with electron and proton radiation was also investigated.

Edwards, David L.↗
NASA NTRS2004

Solar Sail Material Performance Property Response to Space Environmental Effects

The National Aeronautics and Space Administration's (NASA) Marshall Space Flight Center (MSFC) continues research into the utilization of photonic materials for spacecraft propulsion. Spacecraft propulsion, using photonic materials, will be achieved using a solar sail. A solar sail operates on the principle that photons, originating from the sun, impart pressure to the sail and therefore provide a source for spacecraft propulsion. The pressure imparted to a solar sail can be increased, up to a factor of two if the sun-facing surface is perfectly reflective. Therefore, these solar sails are generally composed of a highly reflective metallic sun-facing layer, a thin polymeric substrate and occasionally a highly emissive back surface. Near term solar sail propelled science missions are targeting the Lagrange point 1 (L1) as well as locations sunward of L1 as destinations. These near term missions include the Solar Polar Imager' and the L1 Diamond '. The Environmental Effects Group at NASA's Marshall Space Fliglit Center (MSFC) continues to actively characterize solar sail material in preparation for these near term solar sail missions. Previous investigations indicated that space environmental effects on sail material thermo-optical properties were minimal and would not significantly affect the propulsion efficiency of the sail3-'. These investigations also indicated that the sail material mechanical stability degrades with increasing radiation exposure. This paper will further quantify the effect of space environmental exposure on the mechanical properties of candidate sail materials. Candidate sail materials for these missions include Aluminum coated Mylar TM, Teonexm, and CP1 (Colorless Polyimide). These materials were subjected to uniform radiation doses of electrons and protons in individual exposures sequences. Dose values ranged from 100 Mrads to over 5 Grads. The engineering performance property responses of thermo-optical and mechanical properties were characterized. The contribution of Near Ultraviolet (NUV) radiation combined with electron and proton radiation was also investigated. Conclusions will be presented providing a gauge of measure for engineering performance stability for sails operating in the L1 space environment.

Edwards, David L.↗
NASA NTRS2004

Solar Sail Material Performance Property Response to Space Environmental Effects

The National Aeronautics and Space Administration's (NASA) Marshall Space Flight Center (MSFC) continues research into the utilization of photonic materials for spacecraft propulsion. Spacecraft propulsion, using photonic materials, will be achieved using a solar sail. A solar sail operates on the principle that photons, originating from the sun, impart pressure to the sail and therefore provide a source for spacecraft propulsion. The pressure imparted to a solar sail can be increased, up to a factor of two, if the sun-facing surface is perfectly reflective. Therefore, these solar sails are generally composed of a highly reflective metallic sun-facing layer, a thin polymeric substrate and occasionally a highly emissive back surface. Near term solar sail propelled science missions are targeting the Lagrange point 1 (Ll) as well as locations sunward of L1 as destinations. These near term missions include the Solar Polar Imager and the L1 Diamond. The Environmental Effects Group at NASA s Marshall Space Flight Center (MSFC) continues to actively characterize solar sail material in preparation for these near term solar sail missions. Previous investigations indicated that space environmental effects on sail material thermo-optical properties were minimal and would not significantly affect the propulsion efficiency of the sail. These investigations also indicated that the sail material mechanical stability degrades with increasing radiation exposure. This paper will further quantify the effect of space environmental exposure on the mechanical properties of candidate sail materials. Candidate sail materials for these missions include Aluminum coated Mylar[TM], Teonex[TM], and CPl (Colorless Polyimide). These materials were subjected to uniform radiation doses of electrons and protons in individual exposures sequences. Dose values ranged from 100 Mrads to over 5 Grads. The engineering performance property responses of thermo-optical and mechanical properties were characterized. The contribution of Near Ultraviolet (NUV) radiation combined with electron and proton radiation was also investigated.

Edwards, David L.↗
NASA NTRS2004

Status of Solar Sail Material Characterization at NASA's Marshall Space Flight Center

Near term solar sail propelled science missions are targeting the Lagrange point 1 (Ll) as well as locations sunward of L1 as destinations. These near term missions include the Solar Polar Imager' and the L1 Diamond '. The Environmental Effects Group at NASA's Marshall Space Flight Center (MSFC) continues to actively characterize solar sail material in preparation for these near term solar sail missions. Previous investigations indicated that space environmental effects on sail material thermo-optical properties were minimal and would not significantly affect the propulsion efficiency of the sail. These investigations also indicated that the sail material mechanical stability degrades with increasing radiation exposure. This paper will further quantify the effect of space environmental exposure on the mechanical properties of candidate sail materials. Candidate sail materials for these missions include Aluminum coated Mylar(TradeMark), Teonex(TraeMark), and CP1 (Colorless Polyimide). Experimental data will be presented on sail material response to charged particle radiation and subsequent Hypervelocity Impact (HVI). Data will also be presented indicating mechanical property variations in sail material resulting from electron exposure, proton exposure, and a combined electron and proton exposure. Tabular data consisting of areal density, thickness, thermo-optical, mechanical, and electrical properties, vacuum stability and outgassing will be presented.

Edwards, David L.↗
NASA NTRS2009

Precise Stabilization of the Optical Frequency of WGMRs

Crystalline whispering gallery mode resonators (CWGMRs) made of crystals with axial symmetry have ordinary and extraordinary families of optical modes. These modes have substantially different thermo-refractive constants. This results in a very sharp dependence of differential detuning of optical frequency on effective temperature. This frequency difference compared with clock gives an error signal for precise compensation of the random fluctuations of optical frequency. Certain crystals, like MgF2, have turnover points where the thermo-refractive effect is completely nullified. An advantage for applications using WGMRs for frequency stabilization is in the possibility of manufacturing resonators out of practically any optically transparent crystal. It is known that there are crystals with negative and zero thermal expansion at some specific temperatures. Doping changes properties of the crystals and it is possible to create an optically transparent crystal with zero thermal expansion at room temperature. With this innovation s stabilization technique, the resultant WGMR will have absolute frequency stability The expansion of the resonator s body can be completely compensated for by nonlinear elements. This results in compensation of linear thermal expansion (see figure). In three-mode, the MgF2 resonator, if tuned at the turnover thermal point, can compensate for all types of random thermal-related frequency drift. Simplified dual-mode method is also available. This creates miniature optical resonators with good short- and long-term stability for passive secondary frequency ethalon and an active resonator for active secondary frequency standard (a narrowband laser with long-term stability).

Savchenkov, Anatoliy↗
NASA NTRS2003

Space Durable Polyimide/Carbon Nanotube Composite Films for Electrostatic Charge Mitigation

Low color, space environmentally durable polymeric films with sufficient electrical conductivity to mitigate electrostatic charge (ESC) build-up have been under investigation as part of a materials development activity. These materials have potential applications on advanced spacecraft, particularly on large, deployable, ultra-light weight Gossamer spacecraft. The approach taken to impart sufficient electrical conductivity into the polymer film is based on the use of single walled carbon nanotubes (SWNT) as conductive additives. Earlier approaches investigated in our lab involved both an in-situ polymerization approach and addition of SWNT to an oligomer containing reactive end-groups as methods to improve SWNT dispersion. The work described herein is based on the spray coating of a SWNT/solvent dispersion onto the film surface. Two types of polyimides were investigated, one with reactive end groups that can lead to bond formation between the oligomer chain and the SWNT surface and those without reactive end-groups. Surface conductivities (measured as surface resistance) in the range sufficient for ESC mitigation were achieved with minimal effects on the mechanical, optical, thermo-optical properties of the film as compared to the other methods. The chemistry and physical properties of these nanocomposites will be discussed.

Watson, Kent A.↗
NASA NTRS2003

Electron Radiation Effects on Candidate Solar Sail Material

Solar sailing is a unique form of propulsion where a spacecraft gains momentum from incident photons. Solar sails are not limited by reaction mass and provide continual acceleration, reduced only by the lifetime of the lightweight film in the space environment and the distance to the Sun. Once thought to be difficult or impossible, solar sailing has come out of science fiction and into the realm of possibility. Any spacecraft using this propulsion method would need to deploy a thin sail that could be as large as many kilometers in extent. The availability of strong, ultra lightweight, and radiation resistant materials will determine the future of solar sailing. The National Aeronautics and Space Administration's (NASA) Marshall Space Flight Center (MSFC) is concentrating research into the utilization of ultra lightweight materials for spacecraft propulsion. The Space Environmental Effects Team at MSFC is actively characterizing candidate solar sail material to evaluate the thermo-optical and mechanical properties after exposure to space environmental effects. This paper will describe the irradiation of candidate solar sail materials to energetic electrons, in vacuum, to determine the hardness of several candidate sail materials.

Edwards, David L.↗
NASA NTRS2003

Characterization of Candidate Solar Sail Material Exposed to Space Environmental Effects

Solar sailing is a unique form of propulsion where a spacecraft gains momentum from incident photons. Solar sails are not limited by reaction mass and provide continual acceleration, reduced only by the lifetime of the lightweight film in the space environment and the distance to the Sun. Once thought to be difficult or impossible, solar sailing has come out of science fiction and into the realm of possibility. Any spacecraft using this method would need to deploy a thin sail that could be as large as many kilometers in extent. The availability of strong, ultra lightweight, and radiation resistant materials will determine the future of solar sailing. The National Aeronautics and Space Administration's Marshall Space Flight Center (MSFC) is concentrating research into the utilization of ultra lightweight materials for spacecraft propulsion. The Space Environmental Effects Team at MSFC is actively characterizing candidate solar sail material to evaluate the thermo-optical and mechanical properties after exposure to space environmental effects. This paper will describe the exposure of candidate solar sail materials to emulated space environmental effects including energetic electrons, combined electrons and Ultraviolet radiation, and hypervelocity impact of irradiated solar sail material. This paper will describe the testing procedure and the material characterization results of this investigation.

Edwards, David↗
NASA NTRS2004

Polyimide/Carbon Nanotube Composite Films for Electrostatic Charge Mitigation

Low color, space environmentally durable polymeric films with sufficient electrical conductivity to mitigate electrostatic charge (ESC) build-up have potential applications on large, deployable, ultra-light weight Gossamer spacecraft as thin film membranes on antennas, solar sails, thermal/optical coatings, multi-layer insulation blankets, etc.. The challenge has been to develop a method to impart robust electrical conductivity into these materials without increasing solar absorptivity (alpha ) or decreasing optical transparency or film flexibility. Since these spacecraft will require significant compaction prior to launch, the film portion of the spacecraft will require folding. The state-of-the-art clear, conductive coating (e.g. indium-tin-oxide, ITO) is brittle and cannot tolerate folding. In this report, doping a polymer with single-walled carbon nanotubes (SWNTs) using two different methods afforded materials with good flexibility and surface conductivities in the range sufficient for ESC mitigation. A coating method afforded materials with minimal effects on the mechanical, optical, and thermo-optical properties as compared to dispersal of SWNTs in the matrix. The chemistry and physical properties of these nanocomposites are discussed.

Smith, Joseph G., Jr.↗
NASA NTRS1981

Materials characterization study of conductive flexible second surface mirrors

The status of prequalification and qualification work on conductive flexible second surface mirrors is described. The basic material is FEP Teflon witn either aluminium or silver vacuum deposited reflectors. The top layer has been made conductive by deposition of layer of a indium oxide. The results of a prequalification program comprised of decontamination, humidity, thermal cycling, thermal shock and vibration tests are presented. Thermo-optical and electrical properties. The results of a prequalification program comprised of decontamination, humidity, thermal cycling, thermal shock and vibration tests are presented. Thermo-optical and electrical properties, the electrostatic behavior of the materials under simulated substorm environment and electrical conductivity at low temperatures are characterized. The effects of simulated ultra violet and particles irradiation on electrical and thermo-optical properties of the materials are also presented.

Levadou, F.↗
NASA NTRS2013

A Hybrid Fiber/Solid-State Regenerative Amplifier with Tunable Pulse Widths for Satellite Laser Ranging

A fiber/solid-state hybrid seeded regenerative amplifier, capable of achieving high output energy with tunable pulse widths, has been developed for satellite laser ranging applications. The regenerative amplifier cavity uses a pair of Nd:YAG zigzag slabs oriented orthogonally to one another in order to make thermal lensing effects symmetrical and simplify optical correction schemes. The seed laser used is a fiber-coupled 1,064-nm narrowband (<0.02 nm) diode laser that is discretely driven in a new short-pulsed mode, enabling continuously tunable seed pulse widths in the 0.2-to-0.4-ns range. The amplifier gain unit consists of a pair of Brewster-cut 6-bounce zigzag Nd:YAG laser slabs, oriented 90deg relative to each other in the amplifier head. This arrangement creates a net-symmetrical thermal lens effect (an opposing singleaxis effect in each slab), and makes thermo-optical corrections simple by optimizing the curvature of the nearest cavity mirror. Each slab is pumped by a single 120-W, pulsed 808-nm laser diode array. In this configuration, the average pump beam distribution in the slabs had a 1-D Gaussian shape, which matches the estimated cavity mode size. A half-wave plate between the slabs reduces losses from Fresnel reflections due to the orthogonal slabs Brewster-cut end faces. Successful "temporal" seeding of the regenerative amplifier cavity results in a cavity Q-switch pulse envelope segmenting into shorter pulses, each having the width of the input seed, and having a uniform temporal separation corresponding to the cavity round-trip time of approx. =10 ns. The pulse energy is allowed to build on successive passes in the regenerative amplifier cavity until a maximum is reached, (when cavity gains and losses are equal), after which the pulse is electro- optically switched out on the next round trip The overall gain of the amplifier is approx. =82 dB (or a factor of 1.26 million). After directing the amplified output through a LBO frequency doubling crystal, approx. = 2.1 W of 532-nm output (>1 mJ) was measured. This corresponds to a nonlinear conversion efficiency of >60%. Furthermore, by pulse pumping this system, a single pulse per laser shot can be created for the SLR (satellite laser ranging) measurement, and this can be ejected into the instrument. This is operated at the precise frequency needed by the measurement, as opposed to commercial short-pulsed, mode-locked systems that need to operate in a continuous fashion, or CW (continuous wave), and create pulses at many MHz. Therefore, this design does not need to throw away or dump 99% of the laser energy to produce what is required; this system can be far smaller, more efficient, cheaper, and readily deployed in the field when packaged efficiently. Finally, by producing custom diode seed pulses electronically, two major advantages over commercial systems are realized: First, this pulse shape is customizable and not affected by the cavity length or gain of the amplifier cavity, and second, it can produce adjustable (selectable) pulse widths by simply adding multiple seed diodes and coupling each into commercial, low-cost fiber-optic combiners.

Coyle, Barry↗
NASA NTRS1984

Thermal control coatings experiment (A0138-6)

The objectives of this LDEF experiment are to verify the validity of space environment simulation performed in the laboratory in order to measure the stability of the thermo-optical properties of thermal control coatings, and to compare the behavior in space of some materials for which the available ultraviolet solar simulation is inadequate (especially in the far ultraviolet). The experimental approach is to passively expose samples of the thermal coatings of interest. These coatings include black paint, aluminum paint, white paint, a solar absorber, an optical surface reflector, second surface mirrors, metal coatings, and silica fabrics. Preflight and postflight measurements of thermo-optical properties will be compared to determine the effects of space environment exposure.

Paillous, A.↗
NASA NTRS2006

Preparation and Properties of Nanocomposites Prepared From Shortened, Functionalized Single-Walled Carbon Nanotubes

As part of a continuing materials development activity, low color space environmentally stable polymeric materials that possess sufficient electrical conductivity for electrostatic charge dissipation (ESD) have been investigated. One method of incorporating sufficient electrical conductivity for ESD without detrimental effects on other polymer properties of interest (i.e., optical and thermo-optical) is through the incorporation of single-walled carbon nanotubes (SWNTs). However, SWNTs are difficult to fully disperse in the polymer matrix. One means of improving dispersion is by shortening and functionalizing SWNTs. While this improves dispersion, other properties (i.e., electrical) of the SWNTs can be affected which can in turn alter the final nanocomposite properties. Additionally, functionalization of the polymer matrix can also influence nanocomposite properties obtained from shortened, functionalized SWNTs. The preparation and characterization of nanocomposites fabricated from a polyimide, both functionalized and unfunctionalized, and shortened, functionalized SWNTs will be presented.

Smith, J. G., Jr.↗
NASA NTRS2003

Thermo-Optical and Mechanical Property Testing of Candidate Solar Sail Materials

Solar sailing is a unique form of propulsion where a spacecraft gains momentum from incident photons. Since sails are not limited by reaction mass, they provide continual acceleration, reduced only by the lifetime of the lightweight film in the space environment and the distance to the Sun. Practical solar sails can expand the number of possible missions, enabling new concepts that are difficult by conventional means. The National Aeronautics and Space Administration's Marshall Space Flight Center (MSFC) is concentrating research into the utilization of ultra-lightweight materials for spacecraft propulsion. Solar sails are generally composed of a highly reflective metallic front layer, a thin polymeric substrate, and occasionally a highly emissive back surface. The Space Environmental Effects Team at MSFC is actively characterizing candidate sails to evaluate the thermo-optical and mechanical properties after exposure to electrons. This poster will discuss the preliminary results of this research.

Hollerman, WIlliam A.↗
NASA NTRS2008

Developing Optimized Trajectories Derived from Mission and Thermo-Structural Constraints

In conjunction with NASA and the Department of Defense, the Johns Hopkins University Applied Physics Laboratory (JHU/APL) has been investigating analytical techniques to address many of the fundamental issues associated with solar exploration spacecraft and high-speed atmospheric vehicle systems. These issues include: thermo-structural response including the effects of thermal management via the use of surface optical properties for high-temperature composite structures; aerodynamics with the effects of non-equilibrium chemistry and gas radiation; and aero-thermodynamics with the effects of material ablation for a wide range of thermal protection system (TPS) materials. The need exists to integrate these discrete tools into a common framework that enables the investigation of interdisciplinary interactions (including analysis tool, applied load, and environment uncertainties) to provide high fidelity solutions. In addition to developing robust tools for the coupling of aerodynamically induced thermal and mechanical loads, JHU/APL has been studying the optimal design of high-speed vehicles as a function of their trajectory. Under traditional design methodology the optimization of system level mission parameters such as range and time of flight is performed independently of the optimization for thermal and mechanical constraints such as stress and temperature. A truly optimal trajectory should optimize over the entire range of mission and thermo-mechanical constraints. Under this research, a framework for the robust analysis of high-speed spacecraft and atmospheric vehicle systems has been developed. It has been built around a generic, loosely coupled framework such that a variety of readily available analysis tools can be used. The methodology immediately addresses many of the current analysis inadequacies and allows for future extension in order to handle more complex problems.

Lear, Matthew H.↗